A233307 a(n) = |{0 < k < n: p(k)^2 + q(n-k)^2 is prime}|, where p(.) is the partition function (A000041) and q(.) is the strict partition function (A000009).

0, 1, 2, 2, 1, 1, 4, 2, 3, 2, 2, 4, 4, 3, 2, 2, 5, 3, 1, 5, 3, 5, 6, 3, 3, 2, 2, 1, 1, 2, 2, 5, 3, 4, 3, 5, 3, 1, 6, 4, 7, 10, 3, 5, 4, 2, 4, 5, 3, 4, 2, 3, 7, 9, 5, 6, 8, 2, 5, 3, 3, 5, 4, 3, 5, 4, 6, 7, 6, 3, 2, 9, 8, 6, 1, 6, 7, 7, 6, 2, 5, 8, 4, 6, 2, 6, 4, 8, 7, 3, 5, 3, 3, 5, 4, 5, 8, 5, 6, 2

Offset

1,3

Comments

Conjecture: (i) a(n) > 0 for all n > 1. Also, for any integer n > 4, p(k)*q(n-k) - 1 is prime for some 0 < k < n/2.

(ii) If n > 9, then prime(k)*p(n-k) + 1 is prime for some 0 < k < n. If n > 2, then prime(k)*q(n-k) - 1 is prime for some 0 < k < n, and also prime(k)*q(n-k) + 1 is prime for some 0 < k < n.

(iii) If n > 11, then prime(k) + p(n-k) is prime for some 0 < k < n. If n > 4, then prime(k) + q(n-k) is prime for some 0 < k < n, and also prime(k)^2 + q(n-k)^2 is prime for some 0 < k < n.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Z.-W. Sun, On a^n+ bn modulo m, arXiv preprint arXiv:1312.1166 [math.NT], 2013-2014.

Example

a(5) = 1 since 5 = 1 + 4 with p(1)^2 + q(4)^2 = 1^2 + 2^2 = 5 prime.
a(6) = 1 since 6 = 3 + 3 with p(3)^2 + q(3)^2 = 3^2 + 2^2 = 13 prime.
a(19) = 1 since 19 = 3 + 16 with p(3)^2 + q(16)^2 = 3^2 + 32^2 = 1033 prime.
a(28) = 1 since 28 = 3 + 25 with p(3)^2 + q(25)^2 = 3^2 + 142^2 = 20173 prime.
a(29) = 1 since 29 = 6 + 23 with p(6)^2 + q(23)^2 = 11^2 + 104^2 = 10937 prime.
a(38) = 1 since 38 = 1 + 37 with p(1)^2 + q(37)^2 = 1^2 + 760^2 = 577601 prime.
a(75) = 1 since 75 = 13 + 62 with p(13)^2 + q(62)^2 = 101^2 + 13394^2 = 179409437 prime.
a(160) = 1 since 160 = 48 + 112 with p(48)^2 + q(112)^2 = 147273^2 + 1177438^2 = 1408049580373 prime.
a(210) = 1 since 210 = 71 + 139 with p(71)^2 + q(139)^2 = 4697205^2 + 8953856^2 = 102235272080761 prime.

Mathematica

a[n_]:=Sum[If[PrimeQ[PartitionsP[k]^2+PartitionsQ[n-k]^2], 1, 0], {k, 1, n-1}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A000009, A000040, A000041, A231201, A232504.

Sequence in context: A006694 A210481 A217209 * A202735 A343862 A116595

Adjacent sequences: A233304 A233305 A233306 * A233308 A233309 A233310

Keyword

nonn

Author

Zhi-Wei Sun, Dec 07 2013

Status

approved